Finite Group Databasebeta

← #36355531

Subgroups — #36355531

Order 22144.

216 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012293C211yesmaximal, characteristic
2C2×#19149622C211yesmaximal, characteristic, complemented
3#26011763C211yesmaximal, characteristic, complemented
4#26011305C211yesmaximal, characteristic, complemented
5#26011298C211yesmaximal, characteristic, complemented
6#26011276C211yesmaximal, characteristic, complemented
7#510×C173C211yesmaximal, characteristic, complemented
8#19149725—21no—
9#19149710—21no—
10#19149706—22no—
11#19149704—21no—
12#19149702—21no—
13C2×D1384C2211yescharacteristic
14D8×D173—21no—
15#19149685C2211yescharacteristic
16#19149622—22no—
17C2×#14583801C2211yescharacteristic, complemented
18#19149615C2211yescharacteristic, complemented
19#19149598—21no—
20#135×C173C2211yescharacteristic
21C2×Q16×C173C2211yescharacteristic, complemented
22Q32×C173—21no—
23SD32×C173—21no—
24M32×C173C2211yescharacteristic, complemented
25#14583827—22no—
26#14583826—41no—
27#14583826D411yescharacteristic
28C2×D692—21no—
29C2×D692—41no—
30D4×D173—22no—
31D4×D173—42no—
32#14583816—41no—
33#14583813—22no—
34#14583813—41no—
35D1384—21no—
36D1384D412yes—
37#14583811—22no—
38#14583810—21no—
39#14583810D411yescharacteristic
40#14583809—21no—
41C8×D173—21no—
42#14583801—21no—
43#41×C173—21no—
44C2×Q8×C173—21no—
45Q16×C173—21no—
46Q16×C173—21no—
47Q16×C173D411yescharacteristic
48SD16×C173—21no—
49D8×C173D411yescharacteristic
50C2×C8×C173C2311yescharacteristic
51C16×C173—21no—
52C22×D173—42no—
53C22×D173—41no—
54Dic346D811yescharacteristic
55D692—22no—
56D692—42no—
57D692—41no—
58D692D811yescharacteristic
59C22⋊D173—21no—
60C22⋊D173—42no—
61C4×D173—21no—
62C4×D173—42no—
63#13063436—21no—
64#13063436—41no—
65Q8×C173—21no—
66Q8×C173—41no—
67Q8×C173—41no—
68D4×C173—21no—
69D4×C173—41no—
70C2×C4×C173—41no—
71C2×C4×C173—41no—
72C2×C4×C173C2×D411yescharacteristic
73C8×C173C2×D411yescharacteristic
74C8×C173C2×D411yescharacteristic
75C22×C173—41no—
76C22×C173AGL(1,ℤ/8)11yescharacteristic
77C2×D173—21no—
78C2×D173—42no—
79C2×D173—162no—
80C2×D173—161no—
81C4×C173—41no—
82C4×C173—81no—
83C4×C173#12011yescharacteristic
84C4×C173C2×D811yescharacteristic
85Dic173—21no—
86C2×C173—21no—
87C2×C173—81no—
88C2×C173#44711yescharacteristic
89D173—41no—
90D173—82no—
91C173#219411yescharacteristic, complemented, Sylow
92#2194—1731nomaximal, Sylow
93#576—1731no—
94#510D17311yescharacteristic, complemented
95C2×SD32—1731no—
96#473—1731no—
97#361—1731no—
98#359—1731no—
99#350—1731no—
100H(5,𝔽2)—3461no—
101#137—3461no—
102AGL(1,ℤ/8)—3462no—
103AGL(1,ℤ/8)—3461no—
104#135—3461no—
105#135C2×D17311yescharacteristic
106C2×Q16C2×D17311yescharacteristic, complemented
107C2×D8—3461no—
108C2×D8—1731no—
109#131—1731no—
110Q32—21no—
111SD32—21no—
112SD32—3462no—
113M32C2×D17311yescharacteristic, complemented
114C2×C16—1731no—
115#108—1731no—
116#101—3461no—
117#41—21no—
118#41—3462no—
119#41—1731no—
120#41—6921no—
121C2×Q8—21no—
122C2×D4—3462no—
123C2×D4—3461no—
124C2×D4—6921no—
125C2×D4—6922no—
126Q16—21no—
127Q16—21no—
128Q16—6921no—
129Q16C22⋊D17311yescharacteristic
130SD16—21no—
131SD16—3462no—
132SD16—6921no—
133D8—3462no—
134D8—3461no—
135D8—1732no—
136D8C22⋊D17311yescharacteristic
137M16—3461no—
138M16—3461no—
139M16—1731no—
140C2×C8—3461no—
141C2×C8C22×D17311yescharacteristic
142C16—21no—
143C16—3461no—
144C23—6922no—
145C23—6921no—
146Q8—21no—
147Q8—41no—
148Q8—41no—
149Q8—1731no—
150D4—21no—
151D4—41no—
152D4—3461no—
153D4—3462no—
154D4—1731no—
155D4—6922no—
156D4—6922no—
157D4—6921no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—3461no—
161C2×C4—6922no—
162C2×C4D4×D17311yescharacteristic
163C8—3461no—
164C8—6921no—
165C8D4×D17311yescharacteristic
166C8C2×C22⋊D17311yescharacteristic
167C22—41no—
168C22—3461no—
169C22—6922no—
170C22—27681no—
171C22—27682no—
172C22#1914970311yescharacteristic
173C4—41no—
174C4—81no—
175C4—3461no—
176C4#1914964711yescharacteristic
177C4D8×D17311yescharacteristic
178C2—21no—
179C2—81no—
180C2—13842no—
181C2—6921no—
182C2#2601162311yescharacteristic
subgroupG/H
centerC2#26011623
commutator, cosocleC2×C8×C173C23
FrattiniC2×C8C22×D173
Fitting#510×C173C2
socleC2×C173#447
prime radical#26012293C2
hypercenter#510D173
hypocenterC173#2194
LCSG ▷ C2×C8×C173 ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#19149622C2semidirect
3. #26011763C2semidirect
4. #26011305C2semidirect
5. #26011298C2semidirect
6. #26011276C2semidirect
7. #510×C173C2semidirect
17. C2×#14583801C22semidirect
18. #19149615C22semidirect
21. C2×Q16×C173C22semidirect
24. M32×C173C22semidirect
91. C173#2194semidirect
94. #510D173semidirect
106. C2×Q16C2×D173semidirect
113. M32C2×D173semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#260122932
C2×#191496222
#260117632
#260113052
#260112982
#260112762
#510×C1732
#2194173